Sure, here are the requested mathematical expressions in display mode:

\[
\begin{aligned}
P(H \mid E) & =\frac{P(E \mid H) \cdot P(H)}{P(E)} \\
p(\boldsymbol{X} \mid \alpha) & =\int_{\theta} p(\boldsymbol{X} \mid \theta) p(\theta \mid \alpha) \mathrm{d} \theta \\
p(\theta \mid \boldsymbol{X}, \alpha) & =\frac{p(\boldsymbol{X} \mid \theta) p(\theta \mid \alpha)}{p(\boldsymbol{X} \mid \alpha)} \propto p(\boldsymbol{X} \mid \theta) p(\theta \mid \alpha)
\end{aligned}
\]

\[
\begin{array}{l}
\int \tan x d x=-\int \frac{1}{\cos x} d(\cos x)=-\ln |\cos x|+c ; \int \cot d x=\int \frac{d(\sin x)}{\sin x}=\ln |\sin x|+c \\
\int \frac{d x}{\sqrt{a^{2}-x^{2}}}=\\int \frac{1}{\sqrt{1-\left(\frac{x}{a}\right)^{2}}} d\left(\frac{x}{a}\right)=\arcsin \frac{x}{a}+c ; \int \frac{1}{a^{2}+x^{2}} d x=\frac{1}{a} \int \frac{1}{1+\left(\frac{x}{a}\right)^{2}} d\left(\frac{x}{a}\right)=\frac{1}{a} \arctan \frac{x}{a}+c
\end{array}
\]

I'm unable to render the images in this environment. If you have specific equations or parts of the text that you'd like explained further, please let me know!